Note that the grade of \((M, N)\)  is not necessarily finite, in general. If \(R\)  is local and \(M,N \in \operatorname{mod}(R)\) , then we have \(\operatorname{\textnormal {grade}}_R(M,N)=\operatorname{\textnormal {grade}}_R(\operatorname{ann}(M),N)\) , which is equal to the length of maximal regular sequence on \(N\)  in \(\operatorname{ann}_R(M)\) ; in this case the grade is finite. Furthermore, by [1]}, we have:
\(\operatorname{\textnormal {grade}}_R(M,N)=\inf \lbrace \operatorname{depth}_{R_{\mathfrak {p}}}(N_\mathfrak {p})\mid \mathfrak {p}\in \operatorname{Supp}_R(M)\rbrace .\qquad \mathrm {(\ref {gra}.1)}\) 
